R Notebook

1. Installing Packages

# Disabling Warnings
options(warn = -1)

# Required Packages
packages = c('tseries','TSstudio','fBasics','rcompanion', 'forecast', 'lmtest', 'forecast', 'tsDyn', 'vars', 'readxl', 'PerformanceAnalytics', 'vrtest','pracma', 'rmgarch', 'urca', 'FinTS', 'zoo', 'rugarch', 'e1071', 'fGarch')

# Install all Packages with Dependencies
# install.packages(packages, dependencies = TRUE) 

# Load all Packages
lapply(packages, require, character.only = TRUE)

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2. Loading and Preprocessing the Data

equity <- read.csv("./Dataset/Equity_Combined.csv")

Data already combined, cleaned, and pre-processed using Excel.

3. Data at Glance

These are the closing prices of benchmark indices of the largest equity exchange of 5 countries:

1. Brazil - Ibovespa: The Ibovespa is the benchmark index of the B3 (Brasil Bolsa Balcão), which is the main stock exchange in Brazil, consisting of the most traded stocks in the Brazilian market.

2. China - Shanghai Composite Index: The Shanghai Composite tracks all stocks (A-shares and B-shares) listed on the Shanghai Stock Exchange. It is the most commonly used benchmark for the Chinese stock market.

3. Indonesia - Jakarta Composite Index (JCI): The JCI is the main stock market index of the Indonesia Stock Exchange (IDX), tracking the performance of all listed companies.

4. India - NIFTY 50: It is the main index of India’s National Stock Exchange (NSE) which features majority of listed companies in India.

5. Mexico - BMV IPC: This index is the benchmark for the Mexican stock market, consisting of a selection of the most liquid stocks listed on the Bolsa Mexicana de Valores (BMV).

head(equity)

4. Calculating Log Returns of Each Market

# Changing the Date Format to 'yyyy-mm-dd'
equity$Common.Date <- as.Date(equity$Common.Date, format = "%d-%m-%Y")

head(equity)

# Calculate Log Return for entire file
LR_Equity = CalculateReturns(equity, method="log")[-1,]
head(LR_Equity)

5. Testing Stationarity: ADF Test

Null Hypothesis (H0): The data has a unit root, which means it is non-stationary.

Alternative Hypothesis (H1): The data does not have a unit root, which means it is stationary

adf.test(LR_Equity$Brazil, alternative = "stationary")

Warning: p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LR_Equity$Brazil
Dickey-Fuller = -7.8633, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Null Hypothesis is Rejected, Log Return Series for Brazil is Stationary.

adf.test(LR_Equity$Indonesia, alternative = "stationary")

Warning: p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LR_Equity$Indonesia
Dickey-Fuller = -8.9393, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Null Hypothesis is Rejected, Log Return Series for Indonesia is Stationary.

adf.test(LR_Equity$India, alternative = "stationary")

Warning: p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LR_Equity$India
Dickey-Fuller = -8.5792, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Null Hypothesis is Rejected, Log Return Series for India is Stationary.

adf.test(LR_Equity$China, alternative = "stationary")

Warning: p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LR_Equity$China
Dickey-Fuller = -8.4318, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Null Hypothesis is Rejected, Log Return Series for China is Stationary.

adf.test(LR_Equity$Mexico, alternative = "stationary")

Warning: p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  LR_Equity$Mexico
Dickey-Fuller = -8.4528, Lag order = 8, p-value = 0.01
alternative hypothesis: stationary

Null Hypothesis is Rejected, Log Return Series for Mexico is Stationary.

Conclusion: All Log Returns are Stationary.

6. Visualizing Log Returns

ts_plot(data.frame(LR_Equity$Mexico, equity$Common.Date[-1]))

Registered S3 methods overwritten by 'htmltools':
  method               from         
  print.html           tools:rstudio
  print.shiny.tag      tools:rstudio
  print.shiny.tag.list tools:rstudio
Registered S3 method overwritten by 'htmlwidgets':
  method           from         
  print.htmlwidget tools:rstudio

ts_plot(data.frame(LR_Equity$India, equity$Common.Date[-1]))

ts_plot(data.frame(LR_Equity$Brazil, equity$Common.Date[-1]))

ts_plot(data.frame(LR_Equity$China, equity$Common.Date[-1]))

ts_plot(data.frame(LR_Equity$Indonesia, equity$Common.Date[-1]))

Conclusion: The data seems to have Volatility Clustering, i.e. large changes in return are followed by large changes of either sign, but we need to test further.

7. Testing Autocorrelation: Ljung-Box Test

• Null Hypothesis (H0): The time series data does not exhibit significant autocorrelation up to lag k.

• Alternative Hypothesis (H1): The time series data exhibits significant autocorrelation up to lag k.

7.1. Brazil

# Finding optimal lag
VARselect(LR_Equity$Brazil, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     2      1      1      2 

$criteria
                   1             2             3             4             5             6
AIC(n) -8.6961416569 -8.6985418188 -8.6970412173 -8.6948625929 -8.6924779286 -8.6912827726
HQ(n)  -8.6904206132 -8.6899602531 -8.6855991298 -8.6805599834 -8.6753147973 -8.6712591194
SC(n)  -8.6814473735 -8.6765003936 -8.6676526504 -8.6581268842 -8.6483950782 -8.6398527805
FPE(n)  0.0001672298  0.0001668289  0.0001670795  0.0001674439  0.0001678437  0.0001680445
                   7             8             9            10            11            12
AIC(n) -8.6879857505 -8.6873738462 -8.6857015397 -8.6827755181 -8.6799663119 -8.6833899646
HQ(n)  -8.6651015754 -8.6616291492 -8.6570963208 -8.6513097773 -8.6456400493 -8.6462031801
SC(n)  -8.6292086167 -8.6212495706 -8.6122301224 -8.6019569590 -8.5918006112 -8.5878771221
FPE(n)  0.0001685996  0.0001687029  0.0001689854  0.0001694807  0.0001699577  0.0001693771
                  13            14            15
AIC(n) -8.6814228314 -8.6781364407 -8.6750414553
HQ(n)  -8.6413755250 -8.6352286124 -8.6292731051
SC(n)  -8.5785628472 -8.5679293148 -8.5574871876
FPE(n)  0.0001697109  0.0001702699  0.0001707981

Optimal Lag = 2

Box.test(LR_Equity$Brazil, lag = 2)

    Box-Pierce test

data:  LR_Equity$Brazil
X-squared = 3.774, df = 2, p-value = 0.1515

Null Hypothesis can’t be rejected, there can be no significant autocorrelation at lag = 2.

7.2. Indonesia

# Finding optimal lag
VARselect(LR_Equity$Indonesia, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     4      1      1      4 

$criteria
                   1             2             3             4             5             6
AIC(n) -9.707547e+00 -9.704204e+00 -9.703800e+00 -9.707698e+00 -9.7060465449 -9.702743e+00
HQ(n)  -9.701826e+00 -9.695622e+00 -9.692358e+00 -9.693396e+00 -9.6888834136 -9.682720e+00
SC(n)  -9.692853e+00 -9.682162e+00 -9.674411e+00 -9.670962e+00 -9.6619636946 -9.651313e+00
FPE(n)  6.082272e-05  6.102643e-05  6.105109e-05  6.081356e-05  0.0000609141  6.111567e-05
                   7             8             9            10            11            12
AIC(n) -9.699436e+00 -9.697360e+00 -9.702167e+00 -9.699370e+00 -9.696519e+00 -9.703667e+00
HQ(n)  -9.676551e+00 -9.671616e+00 -9.673562e+00 -9.667904e+00 -9.662192e+00 -9.666480e+00
SC(n)  -9.640659e+00 -9.631236e+00 -9.628696e+00 -9.618552e+00 -9.608353e+00 -9.608154e+00
FPE(n)  6.131819e-05  6.144562e-05  6.115103e-05  6.132236e-05  6.149755e-05  6.105961e-05
                  13            14            15
AIC(n) -9.700743e+00 -9.6997657137 -9.697933e+00
HQ(n)  -9.660695e+00 -9.6568578854 -9.652165e+00
SC(n)  -9.597883e+00 -9.5895585878 -9.580379e+00
FPE(n)  6.123852e-05  0.0000612985  6.141108e-05

Optimal Lag = 4

Box.test(LR_Equity$Indonesia, lag = 4)

    Box-Pierce test

data:  LR_Equity$Indonesia
X-squared = 6.9053, df = 4, p-value = 0.141

Null Hypothesis can’t be rejected, there can be no significant autocorrelation at lag = 4.

7.3. India

# Finding optimal lag
VARselect(LR_Equity$India, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     1      1      1      1 

$criteria
                   1             2             3             4             5             6
AIC(n) -9.218895e+00 -9.215781e+00 -9.212748e+00 -9.215515e+00 -9.216971e+00 -9.213678e+00
HQ(n)  -9.213174e+00 -9.207199e+00 -9.201306e+00 -9.201212e+00 -9.199808e+00 -9.193654e+00
SC(n)  -9.204200e+00 -9.193739e+00 -9.183359e+00 -9.178779e+00 -9.172888e+00 -9.162248e+00
FPE(n)  9.914822e-05  9.945746e-05  9.975954e-05  9.948395e-05  9.933921e-05  9.966693e-05
                   7             8             9            10            11            12
AIC(n) -9.2103332603 -9.2073462683 -9.2045324682 -9.2014850493 -9.2008925517 -9.1991548894
HQ(n)  -9.1874490852 -9.1816015713 -9.1759272494 -9.1700193085 -9.1665662891 -9.1619681049
SC(n)  -9.1515561264 -9.1412219927 -9.1310610509 -9.1206664903 -9.1127268510 -9.1036420469
FPE(n)  0.0001000009  0.0001003001  0.0001005828  0.0001008899  0.0001009498  0.0001011255
                  13            14            15
AIC(n) -9.1965675892 -9.1932411739 -9.1912384348
HQ(n)  -9.1565202828 -9.1503333456 -9.1454700847
SC(n)  -9.0937076050 -9.0830340480 -9.0736841672
FPE(n)  0.0001013877  0.0001017257  0.0001019299

Optimal Lag = 1

Box.test(LR_Equity$India, lag = 1)

    Box-Pierce test

data:  LR_Equity$India
X-squared = 1.5163, df = 1, p-value = 0.2182

Null Hypothesis can’t be rejected, there can be no significant autocorrelation at lag = 1.

7.4. China

# Finding optimal lag
VARselect(LR_Equity$China, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     3      1      1      3 

$criteria
                   1             2             3             4             5             6
AIC(n) -9.229989e+00 -9.226964e+00 -9.234669e+00 -9.231530e+00 -9.228438e+00 -9.225111e+00
HQ(n)  -9.224268e+00 -9.218382e+00 -9.223227e+00 -9.217228e+00 -9.211275e+00 -9.205087e+00
SC(n)  -9.215294e+00 -9.204922e+00 -9.205281e+00 -9.194795e+00 -9.184355e+00 -9.173681e+00
FPE(n)  9.805434e-05  9.835141e-05  9.759648e-05  9.790334e-05  9.820658e-05  9.853394e-05
                   7             8             9            10            11            12
AIC(n) -9.223730e+00 -9.2203858818 -9.217603e+00 -9.214262e+00 -9.2161787762 -9.213463e+00
HQ(n)  -9.200845e+00 -9.1946411848 -9.188997e+00 -9.182797e+00 -9.1818525135 -9.176276e+00
SC(n)  -9.164952e+00 -9.1542616062 -9.144131e+00 -9.133444e+00 -9.1280130754 -9.117950e+00
FPE(n)  9.867016e-05  0.0000990007  9.927671e-05  9.960897e-05  0.0000994184  9.968894e-05
                  13            14            15
AIC(n) -9.214654e+00 -9.212748e+00 -9.212860e+00
HQ(n)  -9.174606e+00 -9.169840e+00 -9.167092e+00
SC(n)  -9.111794e+00 -9.102541e+00 -9.095306e+00
FPE(n)  9.957045e-05  9.976061e-05  9.974959e-05

Optimal Lag = 3

Box.test(LR_Equity$China, lag = 3)

    Box-Pierce test

data:  LR_Equity$China
X-squared = 7.6383, df = 3, p-value = 0.05411

Null Hypothesis can’t be rejected, there can be no significant autocorrelation at lag = 3.

7.5. Mexico

# Finding optimal lag 
VARselect(LR_Equity$Mexico, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     1      1      1      1 

$criteria
                   1             2             3             4             5             6
AIC(n) -9.0294129009 -9.0279678990 -9.0246259135 -9.0237324668 -9.0205855971 -9.0173324972
HQ(n)  -9.0236918571 -9.0193863334 -9.0131838260 -9.0094298573 -9.0034224658 -8.9973088440
SC(n)  -9.0147186174 -9.0059264738 -8.9952373466 -8.9869967581 -8.9765027467 -8.9659025051
FPE(n)  0.0001198328  0.0001200061  0.0001204079  0.0001205155  0.0001208954  0.0001212894
                   7             8            9           10            11            12
AIC(n) -9.0143629331 -9.0111028898 -9.011771762 -9.008449191 -9.0067476836 -9.0061946533
HQ(n)  -8.9914787581 -8.9853581928 -8.983166543 -8.976983451 -8.9724214210 -8.9690078687
SC(n)  -8.9555857993 -8.9449786143 -8.938300345 -8.927630632 -8.9185819828 -8.9106818108
FPE(n)  0.0001216501  0.0001220475  0.000121966  0.000122372  0.0001225805  0.0001226485
                  13          14            15
AIC(n) -9.0036532818 -9.00033295 -8.9971028545
HQ(n)  -8.9636059754 -8.95742512 -8.9513345043
SC(n)  -8.9007932976 -8.89012582 -8.8795485868
FPE(n)  0.0001229608  0.00012337  0.0001237694

Optimal Lag = 1

Box.test(LR_Equity$Mexico, lag = 1)

    Box-Pierce test

data:  LR_Equity$Mexico
X-squared = 0.68113, df = 1, p-value = 0.4092

Null Hypothesis can’t be rejected, there can be no significant autocorrelation at lag = 1.

Conclusion: All 5 series are likely to have no significant serial autocorrelation.

8. Finding the ARMA Order

auto.arima(LR_Equity$Brazil)

Series: LR_Equity$Brazil 
ARIMA(0,0,0) with zero mean 

sigma^2 = 0.0001659:  log likelihood = 1797.92
AIC=-3593.85   AICc=-3593.84   BIC=-3589.43

auto.arima(LR_Equity$India)

Series: LR_Equity$India 
ARIMA(1,0,0) with non-zero mean 

Coefficients:
          ar1   mean
      -0.0498  7e-04
s.e.   0.0404  4e-04

sigma^2 = 9.726e-05:  log likelihood = 1962.69
AIC=-3919.37   AICc=-3919.33   BIC=-3906.12

auto.arima(LR_Equity$Indonesia)

Series: LR_Equity$Indonesia 
ARIMA(0,0,0) with zero mean 

sigma^2 = 6.072e-05:  log likelihood = 2106.09
AIC=-4210.18   AICc=-4210.17   BIC=-4205.76

auto.arima(LR_Equity$China)

Series: LR_Equity$China 
ARIMA(0,0,0) with zero mean 

sigma^2 = 9.707e-05:  log likelihood = 1962.28
AIC=-3922.56   AICc=-3922.55   BIC=-3918.14

auto.arima(LR_Equity$Mexico)

Series: LR_Equity$Mexico 
ARIMA(0,0,0) with zero mean 

sigma^2 = 0.0001176:  log likelihood = 1903.39
AIC=-3804.79   AICc=-3804.78   BIC=-3800.37

All series, except India have AR Order = 0. -> There can be Autocorrelation in Indian Stocks.
All series have MA order Order = 0. -> No Autocorrelation in error/residual terms.

Conclusion: This confirms there is no autocorrelation in the error terms of all of the log return series, (bcz MA Order = 0), it may be suitable to apply ARCH/GARCH, if volatility clustering is confirmed.

9. ARIMA Modelling for Indian Stocks

ARIMA_India = arima(LR_Equity$India,order = c(1,0,0))
ARIMA_India

Call:
arima(x = LR_Equity$India, order = c(1, 0, 0))

Coefficients:
          ar1  intercept
      -0.0498      7e-04
s.e.   0.0404      4e-04

sigma^2 estimated as 9.694e-05:  log likelihood = 1962.69,  aic = -3919.37

# Significance of AR and MA
coeftest(ARIMA_India)

z test of coefficients:

             Estimate  Std. Error z value Pr(>|z|)  
ar1       -0.04983399  0.04038180 -1.2341  0.21718  
intercept  0.00073924  0.00038097  1.9404  0.05233 .
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Null Hypothesis (H0): The coefficient of the AR(1) term is zero, meaning the AR(1) term is not significant.

Alternative Hypothesis (H1): The coefficient of the AR(1) term is non-zero, meaning the AR(1) term is significant.

P > 0.05: Null Hypothesis can’t be rejected, this means AR Order of 1 might be non-significant for Indian Stocks.

Conclusion: AR(1) for Indian Stocks not significant, i.e. there is no autocorrelation.

10. Testing Heteroscedasticity: ARCH Test

Null Hypothesis (H0): There is no ARCH effect in the time series data. In other words, the variance of the residuals is constant over time (homoscedasticity).

Alternative Hypothesis (H1): There is an ARCH effect in the time series data. In other words, the variance of the residuals changes over time (heteroscedasticity)

ArchTest(LR_Equity$Brazil)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LR_Equity$Brazil
Chi-squared = 30.955, df = 12, p-value = 0.002001

Null Hypothesis Rejected; there is conditional heteroscedasticity.

ArchTest(LR_Equity$China)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LR_Equity$China
Chi-squared = 43.11, df = 12, p-value = 2.163e-05

Null Hypothesis Rejected; there is conditional heteroscedasticity.

ArchTest(LR_Equity$India)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LR_Equity$India
Chi-squared = 52.56, df = 12, p-value = 4.932e-07

Null Hypothesis Rejected; there is conditional heteroscedasticity.

ArchTest(LR_Equity$Indonesia)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LR_Equity$Indonesia
Chi-squared = 60.271, df = 12, p-value = 2.015e-08

Null Hypothesis Rejected; there is conditional heteroscedasticity.

ArchTest(LR_Equity$Mexico)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  LR_Equity$Mexico
Chi-squared = 9.8561, df = 12, p-value = 0.6286

Null Hypothesis can’t be rejected; Mexico’s returns might not have any conditional heteroscedasticity.

Conclusion: Log returns of all 4, except Mexico has Heteroscedasticity, i.e. error terms doesn’t follow i.i.nd.

11. Checking Autocorrelation in square of residuals

• Null Hypothesis (H0): The time series data does not exhibit significant autocorrelation up to lag k.

• Alternative Hypothesis (H1): The time series data exhibits significant autocorrelation up to lag k.

11.1. Brazil

# Finding optimal lag 
VARselect(LR_Equity$Brazil^2, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     9      5      1      9 

$criteria
                   1             2             3             4             5             6
AIC(n) -1.605313e+01 -1.604981e+01 -1.604805e+01 -1.604491e+01 -1.607183e+01 -1.606848e+01
HQ(n)  -1.604741e+01 -1.604122e+01 -1.603661e+01 -1.603061e+01 -1.605466e+01 -1.604846e+01
SC(n)  -1.603844e+01 -1.602776e+01 -1.601867e+01 -1.600817e+01 -1.602774e+01 -1.601705e+01
FPE(n)  1.067119e-07  1.070676e-07  1.072552e-07  1.075931e-07  1.047358e-07  1.050865e-07
                   7             8             9            10            11            12
AIC(n) -1.606521e+01 -1.606476e+01 -1.607687e+01 -1.607494e+01 -1.607186e+01 -1.606927e+01
HQ(n)  -1.604232e+01 -1.603902e+01 -1.604827e+01 -1.604347e+01 -1.603753e+01 -1.603208e+01
SC(n)  -1.600643e+01 -1.599864e+01 -1.600340e+01 -1.599412e+01 -1.598369e+01 -1.597375e+01
FPE(n)  1.054314e-07  1.054783e-07  1.042090e-07  1.044109e-07  1.047329e-07  1.050047e-07
                  13            14            15
AIC(n) -1.606714e+01 -1.606383e+01 -1.606052e+01
HQ(n)  -1.602709e+01 -1.602092e+01 -1.601476e+01
SC(n)  -1.596428e+01 -1.595362e+01 -1.594297e+01
FPE(n)  1.052289e-07  1.055778e-07  1.059275e-07

Optimal Lag = 9

Box.test(LR_Equity$Brazil^2, lag = 9)

    Box-Pierce test

data:  LR_Equity$Brazil^2
X-squared = 29.835, df = 9, p-value = 0.0004679

Null Hypothesis can be rejected, there is significant autocorrelation at lag = 9.

11.2. Indonesia

# Finding optimal lag 
VARselect(LR_Equity$Indonesia^2, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     3      3      3      3 

$criteria
                   1             2             3             4             5             6
AIC(n) -1.788723e+01 -1.788426e+01 -1.795353e+01 -1.795190e+01 -1.794945e+01 -1.794712e+01
HQ(n)  -1.788151e+01 -1.787568e+01 -1.794209e+01 -1.793760e+01 -1.793228e+01 -1.792709e+01
SC(n)  -1.787254e+01 -1.786222e+01 -1.792414e+01 -1.791516e+01 -1.790536e+01 -1.789569e+01
FPE(n)  1.704803e-08  1.709876e-08  1.595445e-08  1.598048e-08  1.601973e-08  1.605706e-08
                   7             8             9            10            11            12
AIC(n) -1.794381e+01 -1.794223e+01 -1.793933e+01 -1.793608e+01 -1.793279e+01 -1.794927e+01
HQ(n)  -1.792093e+01 -1.791648e+01 -1.791072e+01 -1.790462e+01 -1.789846e+01 -1.791209e+01
SC(n)  -1.788503e+01 -1.787610e+01 -1.786585e+01 -1.785527e+01 -1.784462e+01 -1.785376e+01
FPE(n)  1.611029e-08  1.613578e-08  1.618272e-08  1.623527e-08  1.628888e-08  1.602256e-08
                  13            14            15
AIC(n) -1.794606e+01 -1.794275e+01 -1.794404e+01
HQ(n)  -1.790601e+01 -1.789984e+01 -1.789827e+01
SC(n)  -1.784320e+01 -1.783254e+01 -1.782648e+01
FPE(n)  1.607419e-08  1.612755e-08  1.610677e-08

Optimal Lag = 3

Box.test(LR_Equity$Indonesia^2, lag = 3)

    Box-Pierce test

data:  LR_Equity$Indonesia^2
X-squared = 47.135, df = 3, p-value = 3.253e-10

Null Hypothesis can be rejected, there is significant autocorrelation at lag = 3.

11.3. India

# Finding optimal lag 
VARselect(LR_Equity$India^2, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     8      1      1      8 

$criteria
                   1             2             3             4             5             6
AIC(n) -1.650852e+01 -1.650541e+01 -1.651263e+01 -1.650958e+01 -1.650986e+01 -1.650787e+01
HQ(n)  -1.650280e+01 -1.649682e+01 -1.650119e+01 -1.649528e+01 -1.649269e+01 -1.648785e+01
SC(n)  -1.649382e+01 -1.648336e+01 -1.648324e+01 -1.647285e+01 -1.646577e+01 -1.645644e+01
FPE(n)  6.767721e-08  6.788808e-08  6.739943e-08  6.760504e-08  6.758659e-08  6.772099e-08
                   7             8             9            10            11            12
AIC(n) -1.651556e+01 -1.652140e+01 -1.651904e+01 -1.651653e+01 -1.651319e+01 -1.651098e+01
HQ(n)  -1.649268e+01 -1.649565e+01 -1.649044e+01 -1.648506e+01 -1.647887e+01 -1.647380e+01
SC(n)  -1.645679e+01 -1.645527e+01 -1.644557e+01 -1.643571e+01 -1.642503e+01 -1.641547e+01
FPE(n)  6.720200e-08  6.681112e-08  6.696873e-08  6.713733e-08  6.736184e-08  6.751099e-08
                  13            14            15
AIC(n) -1.650771e+01 -1.650444e+01 -1.650442e+01
HQ(n)  -1.646767e+01 -1.646154e+01 -1.645865e+01
SC(n)  -1.640485e+01 -1.639424e+01 -1.638687e+01
FPE(n)  6.773211e-08  6.795409e-08  6.795575e-08

Optimal Lag = 8

Box.test(LR_Equity$India^2, lag = 8)

    Box-Pierce test

data:  LR_Equity$India^2
X-squared = 59.182, df = 8, p-value = 6.743e-10

Null Hypothesis can be rejected, there is significant autocorrelation at lag = 8.

11.4. China

# Finding optimal lag
VARselect(LR_Equity$China^2, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     2      2      2      2 

$criteria
                   1             2             3             4             5             6
AIC(n) -1.693957e+01 -1.694809e+01 -1.694519e+01 -1.694185e+01 -1.694282e+01 -1.694571e+01
HQ(n)  -1.693385e+01 -1.693950e+01 -1.693374e+01 -1.692755e+01 -1.692566e+01 -1.692568e+01
SC(n)  -1.692488e+01 -1.692604e+01 -1.691580e+01 -1.690511e+01 -1.689874e+01 -1.689428e+01
FPE(n)  4.397810e-08  4.360535e-08  4.373196e-08  4.387811e-08  4.383564e-08  4.370924e-08
                   7             8             9            10            11            12
AIC(n) -1.694288e+01 -1.693985e+01 -1.693681e+01 -1.693348e+01 -1.693182e+01 -1.692874e+01
HQ(n)  -1.692000e+01 -1.691410e+01 -1.690821e+01 -1.690202e+01 -1.689749e+01 -1.689155e+01
SC(n)  -1.688410e+01 -1.687372e+01 -1.686334e+01 -1.685266e+01 -1.684365e+01 -1.683322e+01
FPE(n)  4.383305e-08  4.396618e-08  4.409994e-08  4.424704e-08  4.432085e-08  4.445761e-08
                  13            14            15
AIC(n) -1.692679e+01 -1.692454e+01 -1.692244e+01
HQ(n)  -1.688675e+01 -1.688164e+01 -1.687667e+01
SC(n)  -1.682393e+01 -1.681434e+01 -1.680488e+01
FPE(n)  4.454412e-08  4.464454e-08  4.473879e-08

Optimal Lag = 2

Box.test(LR_Equity$China^2, lag = 2)

    Box-Pierce test

data:  LR_Equity$China^2
X-squared = 43.099, df = 2, p-value = 4.376e-10

Null Hypothesis can be rejected, there is significant autocorrelation at lag = 3.

11.5. Mexico

VARselect(LR_Equity$Mexico^2, lag.max = 15, type = "const")

$selection
AIC(n)  HQ(n)  SC(n) FPE(n) 
     1      1      1      1 

$criteria
                   1             2             3             4             5             6
AIC(n) -1.617729e+01 -1.617448e+01 -1.617218e+01 -1.617039e+01 -1.617530e+01 -1.617216e+01
HQ(n)  -1.617156e+01 -1.616590e+01 -1.616074e+01 -1.615609e+01 -1.615813e+01 -1.615213e+01
SC(n)  -1.616259e+01 -1.615244e+01 -1.614279e+01 -1.613366e+01 -1.613121e+01 -1.612073e+01
FPE(n)  9.425279e-08  9.451743e-08  9.473516e-08  9.490457e-08  9.444058e-08  9.473765e-08
                   7             8             9            10            11            12
AIC(n) -1.616962e+01 -1.616696e+01 -1.616361e+01 -1.616075e+01 -1.615751e+01 -1.615439e+01
HQ(n)  -1.614674e+01 -1.614121e+01 -1.613501e+01 -1.612928e+01 -1.612319e+01 -1.611720e+01
SC(n)  -1.611085e+01 -1.610083e+01 -1.609014e+01 -1.607993e+01 -1.606935e+01 -1.605888e+01
FPE(n)  9.497783e-08  9.523158e-08  9.555069e-08  9.582494e-08  9.613551e-08  9.643640e-08
                  13            14            15
AIC(n) -1.615252e+01 -1.614954e+01 -1.614687e+01
HQ(n)  -1.611247e+01 -1.610663e+01 -1.610111e+01
SC(n)  -1.604966e+01 -1.603933e+01 -1.602932e+01
FPE(n)  9.661728e-08  9.690579e-08  9.716438e-08

Optimal Lag = 1

Box.test(LR_Equity$Mexico^2, lag = 1)

    Box-Pierce test

data:  LR_Equity$Mexico^2
X-squared = 1.6664, df = 1, p-value = 0.1967

Null Hypothesis can’t be rejected, there is no significant autocorrelation at lag = 1.

Conclusion: It confirms again that except Mexican market all other markets exhibit ARCH effects or Volatility Clustering.

12. Volatility Persistence

Alpha (α): This coefficient captures the short-term persistence of volatility. It reflects the impact of recent shocks on current volatility. A high alpha indicates that recent news has a strong influence on volatility.

Beta (β): This coefficient measures the long-term persistence of volatility. It represents the effect of past conditional variances on current variance. A high beta suggests that volatility is highly persistent over time.

Volatility Persistence (α+β): If this sum is close to 1, volatility is very persistent, and the impact of shocks diminishes slowly.

13. GARCH Modelling for Capturing Volatility Clustering and Volatility Persistence

13.1. Brazil

# Specify the GARCH(1,1) model
brazil_spec <- ugarchspec(
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  mean.model = list(armaOrder = c(0, 0), include.mean = TRUE)
)

# Fit the model
brazil_garch_fit <- ugarchfit(spec = brazil_spec, data = LR_Equity$Brazil)

brazil_garch_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000179    0.000469   0.381386  0.70292
omega   0.000000    0.000000   0.000001  1.00000
alpha1  0.019367    0.003585   5.401507  0.00000
beta1   0.979111    0.003638 269.158389  0.00000

Robust Standard Errors:
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.000179    0.000471   0.380289 0.703731
omega   0.000000    0.000001   0.000001 1.000000
alpha1  0.019367    0.006450   3.002695 0.002676
beta1   0.979111    0.007301 134.111088 0.000000

LogLikelihood : 1820.544 

Information Criteria
------------------------------------
                    
Akaike       -5.9267
Bayes        -5.8979
Shibata      -5.9268
Hannan-Quinn -5.9155

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1063  0.7444
Lag[2*(p+q)+(p+q)-1][2]    1.6497  0.3281
Lag[4*(p+q)+(p+q)-1][5]    4.1289  0.2384
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.772 0.18311
Lag[2*(p+q)+(p+q)-1][5]     5.214 0.13684
Lag[4*(p+q)+(p+q)-1][9]    10.097 0.04782
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.01268 0.500 2.000 0.91033
ARCH Lag[5]   7.19467 1.440 1.667 0.03130
ARCH Lag[7]   9.08984 2.315 1.543 0.02982

Nyblom stability test
------------------------------------
Joint Statistic:  108.2057
Individual Statistics:              
mu      0.1240
omega  21.1131
alpha1  0.3347
beta1   0.3500

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     15.94       0.6613
2    30     30.02       0.4131
3    40     32.29       0.7680
4    50     57.72       0.1841


Elapsed time : 0.1228669 

• alpha1 (ARCH term): The estimate is 0.019367 with a p-value of 0.00000. This indicates that the ARCH effect is highly significant (p-value < 0.05).

• beta1 (GARCH term): The estimate is 0.979111 with a p-value of 0.00000. This suggests that the GARCH effect is also highly significant.

• Weighted Ljung-Box Test on Standardized Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05 (0.7444, 0.3281, 0.2384), indicating no significant serial autocorrelation in the standardized residuals.

• Weighted Ljung-Box Test on Standardized Squared Residuals: The p-value for lag 4 is 0.04782, which is slightly below 0.05, indicating GARCH might not have captured volatility clustering completely.

• ARCH LM Tests: Lag 5 and Lag 7 show p-values below 0.05, indicating that there is evidence of ARCH effects remaining at these lags.

• Sign Bias Test: Since all the p-values are above 0.05, this indicates that there are no asymmetries in how positive and negative shocks impact the volatility.

Overall, the model is a good fit, but we can possibly explore other symmetrical models for a better fit.

brazil_forecast = ugarchforecast(brazil_garch_fit, n.ahead = 10)
brazil_forecast

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=1971-09-06]:
        Series    Sigma
T+1  0.0001789 0.007476
T+2  0.0001789 0.007471
T+3  0.0001789 0.007465
T+4  0.0001789 0.007459
T+5  0.0001789 0.007454
T+6  0.0001789 0.007448
T+7  0.0001789 0.007442
T+8  0.0001789 0.007437
T+9  0.0001789 0.007431
T+10 0.0001789 0.007425

plot(brazil_forecast)

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

1

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

3

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

0

13.2. Indonesia

# Specify the GARCH(1,1) model
indonesia_spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),   mean.model = list(armaOrder = c(0, 0), include.mean = TRUE))
# Fit the model
indonesia_garch_fit <- ugarchfit(spec = indonesia_spec, data = LR_Equity$Indonesia)
indonesia_garch_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000322    0.000295   1.0924  0.27466
omega   0.000010    0.000000  82.2945  0.00000
alpha1  0.106284    0.012161   8.7400  0.00000
beta1   0.724521    0.024630  29.4163  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000322    0.000292   1.1033   0.2699
omega   0.000010    0.000000  82.7278   0.0000
alpha1  0.106284    0.019347   5.4937   0.0000
beta1   0.724521    0.042790  16.9318   0.0000

LogLikelihood : 2121.832 

Information Criteria
------------------------------------
                    
Akaike       -6.9097
Bayes        -6.8809
Shibata      -6.9098
Hannan-Quinn -6.8985

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.660  0.1976
Lag[2*(p+q)+(p+q)-1][2]     1.669  0.3242
Lag[4*(p+q)+(p+q)-1][5]     3.722  0.2908
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1022  0.7492
Lag[2*(p+q)+(p+q)-1][5]    1.9602  0.6279
Lag[4*(p+q)+(p+q)-1][9]    2.6418  0.8163
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     2.221 0.500 2.000  0.1361
ARCH Lag[5]     2.500 1.440 1.667  0.3709
ARCH Lag[7]     2.636 2.315 1.543  0.5847

Nyblom stability test
------------------------------------
Joint Statistic:  36.3777
Individual Statistics:              
mu     0.04386
omega  4.85238
alpha1 0.12508
beta1  0.28298

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     23.97      0.19747
2    30     43.04      0.04518
3    40     58.13      0.02497
4    50     55.76      0.23562


Elapsed time : 0.1867142 

• alpha1 (ARCH term): The ARCH effect is highly significant (p-value < 0.05).

• beta1 (GARCH term): The GARCH effect is also highly significant (p-value < 0.05).

• Weighted Ljung-Box Test on Standardized Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating no significant serial autocorrelation in the standardized residuals.

• Weighted Ljung-Box Test on Standardized Squared Residuals: TThe p-values for lags 1, 2, and 4 are all greater than 0.05, indicating that model have captured volatility clustering.

• ARCH LM Tests: All p-values above 0.05, indicating that there are no ARCH effects remaining.

• Sign Bias Test: Since all the p-values are above 0.05, this indicates that there are no asymmetries in how positive and negative shocks impact the volatility.

Overall the model is a good fit.

indonesia_forecast = ugarchforecast(indonesia_garch_fit, n.ahead = 10)
indonesia_forecast

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=1971-09-06]:
       Series    Sigma
T+1  0.000322 0.007265
T+2  0.000322 0.007356
T+3  0.000322 0.007431
T+4  0.000322 0.007492
T+5  0.000322 0.007543
T+6  0.000322 0.007585
T+7  0.000322 0.007620
T+8  0.000322 0.007649
T+9  0.000322 0.007673
T+10 0.000322 0.007692

plot(indonesia_forecast)

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

1

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

3

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

0

13.3. India

# Specify the GARCH(1,1) model 
india_spec <- ugarchspec(
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
  mean.model = list(armaOrder = c(0, 0), include.mean = TRUE)) 
# Fit the model 
india_garch_fit <- ugarchfit(spec = india_spec, data = LR_Equity$India)
india_garch_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.001151    0.000355   3.2460  0.00117
omega   0.000008    0.000001  11.6003  0.00000
alpha1  0.146746    0.020439   7.1797  0.00000
beta1   0.779105    0.025187  30.9322  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.001151    0.000400   2.8807 0.003968
omega   0.000008    0.000001   7.0624 0.000000
alpha1  0.146746    0.031928   4.5961 0.000004
beta1   0.779105    0.039053  19.9500 0.000000

LogLikelihood : 1999.105 

Information Criteria
------------------------------------
                    
Akaike       -6.5093
Bayes        -6.4805
Shibata      -6.5094
Hannan-Quinn -6.4981

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.07896  0.7787
Lag[2*(p+q)+(p+q)-1][2]   0.09705  0.9217
Lag[4*(p+q)+(p+q)-1][5]   1.27605  0.7947
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.510  0.2192
Lag[2*(p+q)+(p+q)-1][5]     2.988  0.4092
Lag[4*(p+q)+(p+q)-1][9]     4.757  0.4660
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.2228 0.500 2.000  0.6369
ARCH Lag[5]    0.4644 1.440 1.667  0.8941
ARCH Lag[7]    0.9722 2.315 1.543  0.9180

Nyblom stability test
------------------------------------
Joint Statistic:  5.5123
Individual Statistics:              
mu     0.08033
omega  2.52913
alpha1 0.27316
beta1  0.55558

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     45.76    0.0005359
2    30     55.96    0.0019130
3    40     64.39    0.0064265
4    50     82.51    0.0019342


Elapsed time : 0.2009952 

• alpha1 (ARCH term): The ARCH effect is highly significant (p-value < 0.05).

• beta1 (GARCH term): The GARCH effect is also highly significant (p-value < 0.05).

• Weighted Ljung-Box Test on Standardized Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating no significant serial autocorrelation in the standardized residuals.

• Weighted Ljung-Box Test on Standardized Squared Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating that model have captured volatility clustering.

• ARCH LM Tests: All p-values above 0.05, indicating that there are no ARCH effects remaining.

• Sign Bias Test: Since all the p-values are above 0.05, this indicates that there are no asymmetries in how positive and negative shocks impact the volatility.

Overall the model is a good fit.

india_forecast = ugarchforecast(india_garch_fit, n.ahead = 10) 
india_forecast

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=1971-09-06]:
       Series    Sigma
T+1  0.001151 0.007464
T+2  0.001151 0.007718
T+3  0.001151 0.007945
T+4  0.001151 0.008150
T+5  0.001151 0.008336
T+6  0.001151 0.008504
T+7  0.001151 0.008657
T+8  0.001151 0.008796
T+9  0.001151 0.008922
T+10 0.001151 0.009038

plot(india_forecast)

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

1

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

3

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

0

13.4. China

# Specify the GARCH(1,1) model  
china_spec <- ugarchspec(   
  variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),   
  mean.model = list(armaOrder = c(0, 0), include.mean = TRUE))  
# Fit the model  
china_garch_fit <- ugarchfit(spec = china_spec, data = LR_Equity$China)
china_garch_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000283    0.000362 -0.78095  0.43483
omega   0.000005    0.000001  7.33762  0.00000
alpha1  0.076467    0.011271  6.78413  0.00000
beta1   0.870066    0.017019 51.12368  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000283    0.000339 -0.83368  0.40446
omega   0.000005    0.000001  5.33203  0.00000
alpha1  0.076467    0.008961  8.53290  0.00000
beta1   0.870066    0.013847 62.83572  0.00000

LogLikelihood : 1986.059 

Information Criteria
------------------------------------
                    
Akaike       -6.4668
Bayes        -6.4379
Shibata      -6.4668
Hannan-Quinn -6.4555

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.02683  0.8699
Lag[2*(p+q)+(p+q)-1][2]   0.09168  0.9255
Lag[4*(p+q)+(p+q)-1][5]   2.66366  0.4719
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      0.354  0.5519
Lag[2*(p+q)+(p+q)-1][5]     1.775  0.6724
Lag[4*(p+q)+(p+q)-1][9]     2.516  0.8354
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.02056 0.500 2.000  0.8860
ARCH Lag[5]   0.98330 1.440 1.667  0.7379
ARCH Lag[7]   1.14322 2.315 1.543  0.8892

Nyblom stability test
------------------------------------
Joint Statistic:  4.5902
Individual Statistics:              
mu     0.07227
omega  0.93555
alpha1 0.14935
beta1  0.16504

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     28.99     0.066139
2    30     45.09     0.028828
3    40     60.08     0.016627
4    50     80.88     0.002796


Elapsed time : 0.2192121 

• alpha1 (ARCH term): The ARCH effect is highly significant (p-value < 0.05).

• beta1 (GARCH term): The GARCH effect is also highly significant (p-value < 0.05).

• Weighted Ljung-Box Test on Standardized Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating no significant serial autocorrelation in the standardized residuals.

• Weighted Ljung-Box Test on Standardized Squared Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating that model have captured volatility clustering.

• ARCH LM Tests: All p-values above 0.05, indicating that there are no ARCH effects remaining.

• Sign Bias Test: Since all the p-values are above 0.05, this indicates that there are no asymmetries in how positive and negative shocks impact the volatility.

Overall the model is a good fit.

{r} india_forecast = ugarchforecast(india_garch_fit, n.ahead = 10) india_forecast}

{r} plot(india_forecast)}

13.5. Mexico

# Specify the GARCH(0,1) model
mexico_spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(0, 1)),   mean.model = list(armaOrder = c(0, 0), include.mean = TRUE))
# Fit the model
mexico_garch_fit <- ugarchfit(spec = mexico_spec, data = LR_Equity$Indonesia)
mexico_garch_fit

• Kept ARCH Order = 0, because there were no ARCH effects in Mexican Stocks.

• beta1 (GARCH term): The GARCH effect is also highly significant (p-value < 0.05).

• Weighted Ljung-Box Test on Standardized Residuals: The p-values for lags 1, 2, and 4 are all greater than 0.05, indicating no significant serial autocorrelation in the standardized residuals.

• Sign Bias Test: Since all the p-values are above 0.05, this indicates that there are no asymmetries in how positive and negative shocks impact the volatility.

Overall the model is a good fit.

# Specify the GARCH(0,1) model
mexico_spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(0, 1)),   mean.model = list(armaOrder = c(0, 0), include.mean = TRUE))
# Fit the model
mexico_garch_fit <- ugarchfit(spec = mexico_spec, data = LR_Equity$Indonesia)
mexico_garch_fit

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(0,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
       Estimate  Std. Error    t value Pr(>|t|)
mu     0.000304    0.000315 9.6552e-01  0.33429
omega  0.000000    0.000000 3.4654e+01  0.00000
beta1  0.993221    0.000030 3.2576e+04  0.00000

Robust Standard Errors:
       Estimate  Std. Error    t value Pr(>|t|)
mu     0.000304    0.000326    0.93354  0.35054
omega  0.000000    0.000000   32.50106  0.00000
beta1  0.993221    0.000351 2828.42254  0.00000

LogLikelihood : 2106.857 

Information Criteria
------------------------------------
                    
Akaike       -6.8641
Bayes        -6.8425
Shibata      -6.8642
Hannan-Quinn -6.8557

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.740  0.1871
Lag[2*(p+q)+(p+q)-1][2]     1.749  0.3081
Lag[4*(p+q)+(p+q)-1][5]     4.371  0.2112
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic   p-value
Lag[1]                      4.147 4.171e-02
Lag[2*(p+q)+(p+q)-1][2]     4.357 6.048e-02
Lag[4*(p+q)+(p+q)-1][5]    31.076 1.573e-08
d.o.f=1

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale   P-Value
ARCH Lag[2]    0.4168 0.500 2.000 5.185e-01
ARCH Lag[4]   33.2484 1.397 1.611 4.156e-09
ARCH Lag[6]   39.3139 2.222 1.500 2.107e-10

Nyblom stability test
------------------------------------
Joint Statistic:  4.5649
Individual Statistics:             
mu    0.06863
omega 2.26480
beta1 0.40678

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         0.846 1.01 1.35
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     22.14       0.2774
2    30     28.45       0.4939
3    40     35.29       0.6400
4    50     43.36       0.7002


Elapsed time : 0.1178162 

mexico_forecast = ugarchforecast(mexico_garch_fit, n.ahead = 10)
mexico_forecast

*------------------------------------*
*       GARCH Model Forecast         *
*------------------------------------*
Model: sGARCH
Horizon: 10
Roll Steps: 0
Out of Sample: 0

0-roll forecast [T0=1971-09-06]:
        Series    Sigma
T+1  0.0003041 0.007595
T+2  0.0003041 0.007595
T+3  0.0003041 0.007595
T+4  0.0003041 0.007595
T+5  0.0003041 0.007595
T+6  0.0003041 0.007594
T+7  0.0003041 0.007594
T+8  0.0003041 0.007594
T+9  0.0003041 0.007594
T+10 0.0003041 0.007594

14. Conclusion

We’ve forecasted the log returns of all the series using GARCH Models. Stock markets of all other countries exhibited volatility clustering, except the Mexican markets.